Symplectic geometry | Theorems in differential geometry

Delzant's theorem

In mathematics, a Delzant polytope is a convex polytope in such for each vertex , exactly edges meet at , and these edges form a collection of vectors that form a -basis of . Delzant's theorem, introduced by Thomas Delzant, classifies effective Hamiltonian torus actions on compact connected symplectic manifolds by the image of the associated moment map, which is a Delzant polytope. The theorem states that there is a bijective correspondence between symplectic toric manifolds (up to torus-equivariant symplectomorphism) and Delzant polytopes -- more precisely, the moment polytope of a symplectic toric manifold is a Delzant polytope, every Delzant polytope is the moment polytope of such a manifold, and any two such manifolds with the equivalent moment polytopes (up to translations) admit a torus-equivariant symplectomorphism between them. (Wikipedia).

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Convex polytope | Moment map | Symplectic manifold